qm.ridge {RXshrink}

Restricted (2-parameter) Maximum Likelihood Shrinkage in Regression

Description

Computes and displays TRACEs for a q-Shaped shrinkage PATH, including the m-Extent of shrinkage along that path, most likely under Normal-distribution theory to yield optimal reducions in MSE Risk. When rank(X-matrix) = p > 2, the most likely solution on the qm.ridge() path will be less likely to have minimal MSE risk than the optimal solution on the p-parameter eff.ridge() path. The Hoerl-Kennard "ordinary" ridge path has Shape q=0 within the qm.ridge() family.

Usage

  qm.ridge(form, data, rscale = 1, Q = "qmse", steps = 20, nq = 21,
              qmax = 5, qmin = -5, omdmin = 9.9e-13)

Arguments

form

A regression formula [y~x1+x2+...] suitable for use with lm().

data

Data frame containing observations on all variables in the formula.

rscale

One of three possible choices (0, 1 or 2) for "rescaling" of variables (after being "centered") to remove all "non-essential" ill-conditioning: 0 implies no rescaling; 1 implies divide each variable by its standard error; 2 implies rescale as in option 1 but re-express answers as in option 0.

Q

Shape parameter that controls the curvature of the shrinkage path through regression-coefficient likelihood space (default = "qmse" implies use the value found most likely to be optimal.) Use Q = 0 to specify Hoerl-Kennard "ordinary" ridge regression.

steps

Number of equally spaced values per unit change along the horizontal m-Extent-of-shrinkage axis for estimates to be calculated and displayed in TRACES (default = 20.)

nq

Number of equally spaced values on the lattice of all possible values for shrinkage q-Shape between the "qmin" and "qmax" parameter settings (default = 21.)

qmax

Maximum allowed q-Shape (default = +5.)

qmin

Minimum allowed q-Shape (default = -5.)

omdmin

Strictly positive minimum value for one-minus-delta (default = 9.9e-013.)

Details

Traditional qm.ridge() paths cannot be overall-optimal when p > 2 because they are restricted to using strictly "monotone" (increasing or decreasing) shrinkage factors. Still, the "best" m-Extent of qm-shrinkage is marked by a vertical dashed-line on all 5-types of qm.ridge() TRACE displays. Compared to OLS estimates, these shrunken estimates have higher likelihood of reduced MSE risk and can be much more stable and reasonable, numerically. On the other hand, because only OLS estimates are guaranteed to be minimax when risk is MATRIX valued (truly multivariate), no guarantee of an actual reduction in MSE Risk is necessarily associated with shrinkage.

Value

An output list object of class qm.ridge:

form

The regression formula specified as the first argument.

data

Name of the data.frame object specified as the second argument.

p

Number of regression predictor variables.

n

Number of complete observations after removal of all missing values.

r2

Numerical value of R-square goodness-of-fit statistic.

s2

Numerical value of the residual mean square estimate of error.

prinstat

Listing of principal statistics.

mx

Matrix containing mean values of X-predictors.

crlqstat

Listing of criteria for maximum likelihood selection of path q-Shape.

qmse

Numerical value of q-Shape most likely to be optimal.

qp

Numerical value of the q-Shape actually used for shrinkage.

coef

Matrix of shrinkage-ridge regression coefficient estimates.

risk

Matrix of MSE risk estimates for fitted coefficients.

exev

Matrix of excess MSE eigenvalues (ordinary least squares minus ridge.)

infd

Matrix of direction cosines for the estimated inferior direction, if any.

spat

Matrix of shrinkage pattern multiplicative delta factors.

mlik

Listing of criteria for maximum likelihood selection of m-Extent-of-shrinkage.

sext

Listing of summary statistics for all m-Extents-of-shrinkage.

mClk

Most-Likely Extent of Shrinkage Observed: best multiple of (1/steps) <= p.

minC

Minimum Observed Value of Normal-theory -2*log(Likelihood-Ratio).

QS

Was a Mesh-Search for Best q-Shape requested? : 1 => Yes, 0 => No.

qML

Computable only when p=rank=2: True Most-Likely q-Shape.

kML

Computable only when p=rank=2: True Most-Likely k-Factor.

dML1

Computable only when p=rank=2: True Most-Likely Delta[1]-Factor.

dML2

Computable only when p=rank=2: True Most-Likely Delta[2]-Factor.

mML

Computable only when p=rank=2: True Most-Likely m-Extent.

Author(s)

Bob Obenchain <[e-mail]>

References

Burr TL, Fry HA. (2005) Biased Regression: The Case for Cautious Application. Technometrics 47, 284-296.

Goldstein M, Smith AFM. (1974) Ridge-type estimators for regression analysis. J. Roy. Stat. Soc. B 36, 284-291. (The 2-parameter shrinkage family.)

Obenchain RL. (1975) Ridge Analysis Following a Preliminary Test of the Shrunken Hypothesis. Technometrics 17, 431-441. doi:10.1080/00401706.1975.10489369

Obenchain RL. (1978) Good and Optimal Ridge Estimators. Annals of Statistics 6, 1111-1121. <doi:10.1214/aos/1176344314>

Obenchain RL. (2022) Efficient Generalized Ridge Regression. Open Statistics 3: 1-18. doi:10.1515/stat-2022-0108 [Best q-Shape when p = 2.]

Obenchain RL. (2022) RXshrink_in_R.PDF RXshrink package vignette-like document, Version 2.3. http://localcontrolstatistics.org

See Also

eff.ridge, correct.signs, MLtrue and RXpredict.

Examples

  data(longley2)
  form <- GNP~GNP.deflator+Unemployed+Armed.Forces+Population+Year+Employed
  rxrobj <- qm.ridge(form, data=longley2)
  rxrobj
  plot(rxrobj)
  str(rxrobj)